Formal Concept Analysis

"In information science, formal concept analysis (FCA) is a principled way of deriving a concept hierarchy or formal ontology from a collection of objects and their properties. Each concept in the hierarchy represents the objects sharing some set of properties; and each sub-concept in the hierarchy represents a subset of the objects (as well as a superset of the properties) in the concepts above it. The term was introduced by Rudolf Wille in 1981, and builds on the mathematical theory of lattices and ordered sets that was developed by Garrett Birkhoff and others in the 1930s."

Formal concept analysis can be used as a qualitative method for data analysis. Since the early beginnings of Formal Concept Analysis in the early 1980s, the research group Formal Concept Analysis at the TU Darmstadt has gained experience from more than 200 projects in which Formal Concept Analysis was applied (as of 2005).

More recently, FCA, as a type of data science discipline, has gained prominence in the field of machine learning. Important insights were gained in the Dagstuhl workshops on the subject of Explainable AI (XAI) and Applied Machine Intelligence (AMI), which took place from March 10th to 12th, 2019.

While a concept in formal logic is reduced to its scope as a single-place predicate, formal concept analysis makes the theory of concepts less abstract again by also considering the content of the concept. Thus, the formal analysis of terms is based on the categories of extension and intension of linguistics and the classical logic of terms.

Clarity of concepts is sought in the sense of Charles S. Peirce's pragmatic maxim by unfolding observable, elementary properties of the subsumed objects. In his later philosophy, Peirce assumed that logical thinking aims at grasping reality through the three steps of concept, judgment and conclusion. Mathematics abstracts logical thinking, develops forms of possible reality and can therefore support rational communication. Against this background, Rudolf Wille defines: "The aim and meaning of formal concept analysis as a mathematical theory of concepts and concept hierarchies is to support the rational communication of people by developing mathematically suitable concept structures that can be activated logically."

The main theorem of FCA is also known as Rudolf Wille's main theorem on conceptual associations. Among other things, it states that every complete lattice is isomorphic to a conceptual lattice.

The formal concept analysis uses the term concept, but not in the formal sense that we will summarize at the end of this elaboration under formal concepts.